We consider a simple graph its corresponding edge ideal I in a polynomial ring R. It is well known that upper bounds for the projective dimension of R/I provide lower bounds for the first non-zero homology group of the graph’s independence complex. Determining upper bounds for the projective dimension of R/I is equivalent to finding lower bounds for the depth of R/I. We discuss such bounds as well as lower bounds for the depth of higher powers of the edge ideal. This is joint work with Susan Morey.